The present invention relates generally to a transversal filter, or finite impulse response filter, (which will be referred to as FIR filter) for realizing a desirable frequency property (characteristic) by convolution integration of a finite number of factors (which will be referred to as FIR factor) and delayed signals, and more particularly to method and apparatus for obtaining (designing) the FIR factor so as to realize the desirable frequency property.
FIR filters have been employed for various systems such as tone controller for the purpose of arbitrarily obtaining a frequency characteristic. The desirable frequency characteristic obtained thereby is generally expressed by an amplitude-frequency characteristic (power spectrum), which does not contain phase information. Thus, the FIR factor is a time function and cannot be obtained directly by means of the inverse (reverse)-Fourier transformation thereof. One known method for obtaining the phase information from a power spectrum is to use the Hilbert transformation, thereby obtaining a FIR factor for realizing a desirable amplitude frequency property as disclosed in a document (EA85-44) published by Acoustic Academic Society and Electric-Acoustic Society, 1985, for example. This method will be briefly described hereinbelow with reference to FIG. 1. The Hilbert transformation is a method for conversion of a real variable function or imaginary variable function into a complex variable function, as known from "Fundamentals of Digital Signal Processing" published by Ohm-Sha, for example. That is, when an amplitude frequency property is as indicated by a solid line 11 in (a) of FIG. 1, assuming the amplitude frequency property as a real variable function, the Hilbert transformation of the real variable function 11 is effected and results in a real variable function as indicated by a solid line 12 in (b) of FIG. 1 wherein a dotted line 13 represents an imaginary variable function. Here, on the contrary, an amplitude frequency property is derived from the complex variable function obtained by the above Hilbert transformation as indicated by a solid line 14 in (c) of FIG. 1. As seen from (c) of FIG. 1, this-obtained amplitude frequency property 14 is different from the desirable frequency property indicated by a dotted line 11 in (c) of FIG. 1. Therefore, on order to reduce the difference therebetween, the desirable frequency property is controlled by a percentage of the difference with respect to the respective frequencies and again Hilbert-transformed to obtain an amplitude frequency property which is in turn compared with the desirable amplitude frequency characteristic. The successive comparison therebetween results in obtaining a complex variable function for a desirable amplitude frequency property usable in practice. A FIR factor can be obtained by the inverse-Fourier transformation of the obtained complex variable function. According to the above-mentioned documents, when the Hilbert transformation is considered as a discrete-time system, the transformation equations are expressed as follows. ##EQU1## where H(k) represents a complex variable function to be obtained and P(m) is a real variable function.
In the Hilbert transformation, the value obtained from the equation (2) can be obtained in advance and stored as a data table.
However, because P(m) in the above equation (1) is varied, operation of sum of products is required for realization of the equation (1), and for performing the Hilbert transformation at the point N, the operation of sum of products is performed at least N.sup.2 times. The N.sup.2 -time operation of sum of products determine the transforming time of the Hilbert transformation and the scale of the transformation-arithmetic unit. Since the transforming time thereof and the unit scale are dependent upon the frequency band and frequency resolution, difficulty is encountered to achieve the reduction thereof.
Furthermore, the second-mentioned document discloses a method for obtaining phase information from the power spectrum wherein a transformation is effected in terms of linear phase to obtain a FIR factor for realizing a desirable frequency property. That is, when the linear phase transformation is considered in a discrete-time system, the transformation equations are expressed as follows. EQU Hr(k)=A(k).cos{-(N-1)/N..pi.k} (3) EQU Hi(k)=A(k).sin{-(N-1)/N..pi.k} (4)
Hr(k) is a real variable function obtained, Hi(k) is an imaginary variable function also obtained, and A(k) represents an amplitude frequency property.
However, in the equations (3) and (4), even if cos{-(N -1)/N..pi.k} and sin{-(N-1)/N..pi.k} can be obtained in advance, multiplication must be made at least N.2 times in the case of performing the linear phase transformation at the point N. The N.2-time multiplication determines the transforming time of the linear phase transformation. The linear-phase transformation time depends upon the frequency band and frequency resolution determined, resulting in difficulty of the reduction thereof.